The price and risk of guaranteed interest rate products

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T HE P RICE AND R ISK OF

G UARANTEED I NTEREST R ATE P RODUCTS by

Ørjan Mekkalg˚ arden Ressem

Thesis

presented for the degree Master of Science (Modeling and Data Analysis)

Faculty of Mathematics and Natural Sciences University of Oslo

May 2009

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Abstract

This paper examines the utility indifference price of interest rate products and the risk associated with these. Such products can be compared with put options and are here considered to be written on a non-tradeable asset which can be hedged with a correlated asset. Initially, we look at the case where both the tradeable and non-tradeable assets can be modeled by two geometric Brownian motions. This model is later extended to the case where it is assumed that the tradeable asset follows a L´evy process.

The paper is based on the article ’Utility indifference pricing of interest- rate guarantees’ by Fred Espen Benth and Frank Proske, but is meant to be an independent paper. The definitions of the utility indifference price and the residual risk remaining after hedging are the same as in their paper.

The residual risk is measured with several different risk measures such as Value at Risk, Conditional Value at Risk and Expected Shortfall. These measures, with others, are closely examined and evaluated.

Numerical examples are included showing that the utility indifference price is lower for negative correlation than for positive and that the price can be even lower if the tradeable asset follows a L´evy process. Thus, if e.g.

life companies can hedge in assets allowing jumps, and that are negatively correlated with their pension fund, they may offer lower prices with prac- tically unaltered measures of risk.

Analysis of the pricing and hedging of interest rate guarantees are not only relevant for life companies, but also for other financial institutions offering investment products where there is a guaranteed least rate of return.

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Preface

This thesis has been prepared in partial fulfillment of the requirements for my master degree at the Department of Mathematics, Faculty of Mathematics and Natural Science at the University of Oslo.

Professor Fred Espen Benth, University of Oslo, has been thesis supervisor while Professor Frank Proske, University of Oslo, has been co-supervisor.

The thesis was written over three months in the spring term of 2009 and was mainly produced independently.

There is a certain number of people I wish to thank in relation to my thesis.

In particular, I would like to thank my thesis supervisor Professor Fred Es- pen Benth for an interesting and relatively applicable subject and for useful comments. I would also like to thank co-supervisor Professor Frank Proske for always showing a sincere interest in my work and for his help with various issues. Their theoretical as well as applied knowledge has generated inspiring and helpful discussions.

Lastly, I would like to thank my fellow students and friends for good spirit and friendly atmosphere.

Oslo, May 2009

Ørjan Mekkalg˚arden Ressem

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Introduction

I will in this paper look at the risk associated with interest rate products when priced with the indifference utility pricing method. This method is a tool to be used if it is dubious to assume that a perfect hedge can be achieved and we need to utilize sub-optimal replication strategies. In other words, this price can be viewed as a substitute for the standard Black & Scholes framework. Reasons for why the Black & Scholes framework can be inadequate is e.g. illiquidity or that contracts simply are not long enough compared to the time perspective of a pension fund.

Life insurance companies often offer pension saving deals with a guaranteed least rate of return to their clients. Of course, the companies aim for a higher rate of return, while the customers are protected against low returns. This offer is equivalent to issuing a put option with strike dependent on the guaranteed rate of return and the time perspective. By issuing such a put option, the companies undertake a risk of having to cover the loss should their investments fail to achieve the guaranteed rate of return.

To take this risk, the the issuer needs to be compensated. This compensation is here decided by the utility indifference price. The utility indifference price is defined at the level where the issuer of an option is indifferent between entering the market by its own or issuing the option and entering the market with the collected premium. These two optimal investment problems are solved using stochastic control theory and the difference between them gives the utility based hedging strategy. The construction and notation will be recognizable with the one in Benth and Proske[6].

Choosing the ’best’ model is always a difficult choice. The optimal model is a model describing the reality close to perfect and, at the same time, has few and easy-to-get parameters. This being said, we will in the earlier chapters use a model with as few parameters as possible in this context, and then try to expand it to better fit the reality in the later chapters. In detail the paper will progress as follows.

The first chapter will introduce a model for the financial market of which we are operating in and then use it to define the utility indifference price of a put option and the risk of issuing these. It is all dependent on the risk aversion of the issuer. The lowest price of which a life company is willing to issue such guarantees is obtained when the issuer is indifferent to risk.

In the second chapter, I will look at different risk measures, including Value at Risk, Conditional Value at Risk and Expected Shortfall. I find it difficult discussing the topic of risk without mentioning the highly interesting changes being made in the Basel (worldwide) and Solvency (EU) frameworks, namely the newly implemented Basel II and the soon-to-be implemented Solvency II directives. These will be mentioned briefly.

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Chapter three will be a continuation of both chapter one and chapter two and will contain a risk analysis of the option defined in chapter one.

In chapter four the model in chapter one will extended to include the possibility of investing in a stock behaving like a L´evy process. The implied changes in price and risk will be studied.

The fifth chapter will contain a conclusion of the findings in the paper together with a survey of some of the limitations of the model and some possible im- provements and extensions.

Assumptions

We are assuming an incomplete market, meaning that we cannot hedge any claims perfectly. In other words, there exist no explicit arbitrage-free price, but rather a continuum of such. With the indifference utility pricing method we will find the price within this continuum that is most profitable for a certain agent, i.e. has the highest utility for a given company or investor.

Further, this is a situation where the life company can only use a part of the fund for hedging. Therefor, the fund is interpreted as a non-tradeable asset, whereas the part that can be traded is modeled as a separate correlated asset.

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1 The Model

1.1 Defining the model

In this paper, as in Benth and Proske[6], we are assuming a complete probability space

Ω,F,{F_t}_t∈[0,T], P

supporting two independent Brownian motions B and W. Our algebra of in- formationF_tis assumed to fulfill ’the usual conditions’, namely thatF_tis right continuous and thatF₀ contains all the null sets¹.

We consider a market model consisting of a bank account with deterministic return (a risk free asset), a stock driven byB (a risky asset) and a fund driven by the stock and W (an untradeable asset). The extent of which the fund is driven by both the stock and the Brownian motion W is determined by the correlation parameter, ρ ∈ (−1,1). Note that in order to use the utility indifference pricing method, we have to assume that our market is incomplete, meaning that all claims can not be perfectly hedged, i.e. |ρ| 6= 1. Why this has to hold will be revealed shortly, when we define the utility indifference price.

We assume that the value of the bank account at time t, S_t⁰, the value of the stock at timet,St, and the value of the fund at time t,Yt2, have the following dynamics.

dS_t⁰

S_t⁰ = rdt dS_t

S_t = µdt+σdB_t dY_t

Yt

= νdt+η

ρdB_t+p

1−ρ²dW_t

The parameters, r, µ, ν,σ and η are real constants and the two last ones are positive, by definition since they are volatilities. The three first ones will also be positive in any normal market, but there are situations in which at leastµ andν could be negative; for instance the financial crisis we just witnessed. It is also possible for r to become negative, but this is highly unlikely and belongs only in the field of crisis modeling. This paper will at all times use a positive set of parameters to fit a healthy market.

On the upside, this choice of model assures us that both the stock and the fund is positive at all times. The downside, of course, is that not only will the stock and the fund be positive at all times, they will also be strictly positive at all times. While the former is a property corresponding to the real world, the latter is not; Companies go bankrupt all the time, causing their stocks to loose all

1That is ifB⊂A∈ F0 withP(A) = 0, thenB∈ F0

2It is shown in Benth and Proske thatYt is a Markov process

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value and never recover. This could be modeled by introducing a low threshold b saying that ifSt< b, thenSt= 0 and the company would be bankrupt. The threshold should be chosen such that the probability of S_t assuming a value belowb,P(S_t< b), corresponds to the real market. However, this is not within the topic of this paper, and will not be treated further.

As mentioned in the introduction, the pricing of the fund will be equivalent with the pricing of an European put option with exercise time T and strike K = Y₀e^gT, where g is the guaranteed rate of return of the portfolio. We consider the case when an investor is shortλsuch put options and we letθt be the cash amount invested in S_t while the remaining cash or wealth is invested in the risk free asset, S_t⁰. This gives us the wealth portfolio dynamic:

dX_t^λ,θ^t =θt

dSt

S_t +r(X_t^λ,θ^t −θt)dt (1) where the trading strategy θt is admissible when the equation above has a unique strong solution³,X_t^λ,θ^t, fort∈[0, T] and

E[−U(X_T^λ,θ^t)]<∞ (2) Here U is the utility function our indifference pricing is based on, and is of the form U(x) =−¹_γe^−γx, where γ >0 is the risk aversion of the company. Small γ implies low risk aversion, meaning that the company is willing to take on more risk than if they had a higher γ. At first, one might think that this risk aversion parameter is hard to determine, but, as found in Benth and Proske[6], this parameter can easily be determined from past investments of the company.

This is also mentioned in Benth et al.[7]

As shown in Benth and Proske[6] and the references therein, we can express the utility indifference price as

Definition 1.1.

p^γ_λ(t, y) = e^−r(T^−t)lnw(t, y)

γ(1−ρ²) (3)

Here the function w is of the form w(t, y) = EQ⁰

exp

λγ(1−ρ²)(K−Y_T)⁺ |Y_t=y

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By using thatYtis a Markov process underQ⁰, we can redefine equation (4) to the following equation

w(t, y) = E⁰ h

expn

λγ(1−ρ²)(K−Y_T^t,y))⁺oi

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3As defined in page 72 of Øksendal[2]

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where

Y_T^t,y = y·exp

(δ−1

2η²)(T −t) +η

ρ(B_T⁰ −B_t⁰) +p

1−ρ²(W_T −W_t) This expectation is under the minimal martingale measure Q⁰, under which (B⁰, W) are two independent Brownian motions, with dB_t⁰ = dB_t+ ^µ−r_σ dt.

The fact that B⁰ is a Brownian motion under Q⁰ follows from the Girsanov Theorem. Further, theQ⁰ dynamics ofSt andYt becomes

dSt

St

= rdt+σdW_t⁰ dYt

Y_t = δdt+η

ρdW_t⁰+p

1−ρ²dWt

forδ=ν−ηρ^µ−r_σ .

One should note that since the fund, Yt, is not tradeable, our market is not complete. Hence the utility indifference price, whenρ is tending to ±1, will in general not be equal to the Black and Scholes price.

1.1.1 The utility indifference price under a general martingale measure

Since companies often have a risk aversion not tending to zero, we have to consider the price under some Equivalent Martingale Measure (EMM)Q^γ_λ. We note that this measure is dependent both on the number of put options issued, λ, and the risk aversion factor γ, as it should since the indifference price is nonlinear in both. It is shown in Theorem 3.2 in Benth and Proske[6] that such an EMM exists, making the utility indifference pricep^γ_λ arbitrage free and stating the explicit form of the EMM yielding the utility indifference price. The theorem states the following:

Theorem 1.2. There exists an equivalent martingale measure Q^γ_λ such that p^γ_λ(t, y) = e^−r(T^−t)E_Q^γ

λ

λ(K−Y(T))⁺|Y(t) =y

(6) Moreover, the Q^γ_λ-dynamics ofY_t and S_t are given by

dS_t

S_t = rdt+σdB⁰ (7)

dY_t Yt

= δ^γ(t, Y(t))dt+η

ρdB⁰+p

1−ρ²dW_λ^γ

(8) where (B⁰, W_λ^γ) are two independent Brownian motions underQ^γ_λ, with

dW_λ^γ(t) = dW(t)−1 2ηγp

1−ρ²e^r(T^−t)Y(t)∂_yp^γ_λ(t, Y(t))dt (9) Finally,

δ^γ(t, y) = δ+1

2η²γ(1−ρ²)e^r(T^−t)y∂_yp^γ_λ(t, y) (10)

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Proof. The proof is carried out in full by Benth and Proske showing thatQ^γ_λis a probability measure and then showing that this implies that the representation of the price as a conditional expectation holds.

The indifference pricing measure defined above can be said to be risk neutral in some sense, since the discounted stock price process e^−rtSt is a martingale under it. It is also important to note that even if the utility indifference price is arbitrage free, the market need not be. This is because the price is dependent on the risk aversion, thus making a price that is fair for one trader with risk aversionγ₁ and that is an arbitrage opportunity for another with risk aversion γ2.

Benth and Proske[6] also notes that Q^γ_λ →Q⁰ when γ →0 and that

γ→0limp^γ_λ =p⁰_λ

which are important points saying that the lowest indifference price is obtained when the trader has zero risk aversion. This is perfectly intuitive and easy to verify.

1.1.2 The residual risk

The residual risk is the risk the company is left with after hedging. When we, as in Benth and Proske[6], define the hedge, H_λ^γ, as

H_λ^γ = X_λ^θ^λ−X₀^θ⁰ (11) the residual risk,R^γ_λ, becomes

R^γ_λ =X_λ^θ^λ(T)−X₀^θ⁰(T)−λ(K−Y_T)⁺

that is the payoff of the put option(s) subtracted from the hedge at terminal time. Since X_λ^θ^λ and X₀^θ⁰ is the wealth portfolio when issuing λ and 0 put options, respectively, the risk can be construed as the difference between the wealth portfolio when issuingλput options subtracted the value ofλput options and the value of the wealth portfolio if we did not issue any options. It is shown in section four of Benth and Proske that the residual risk can also be interpreted as the cumulative value of the perfect hedge⁴ with respect to a residual risk process. In other words, it can be represented by

R^γ_λ = Z T

0

e^r(T^−t)Y_t∂_yp^γ_λ(t, Y_t)dR_t (12) Here,dR_t is a residual risk process given by

dR_t = ηρ σ

dS_t S_t −rdt

− dY_t

Y_t −δ^γ(t, Y_t)dt

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4A perfect hedge, a hedge that would eliminate all risk, is only possible in our risk neutral worldQ^γ_λof a complete market

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wheredS_t and dY_t is theQ^γ_λ-dynamics given by equations (7) and (8).

Further, we note that the integrand in equation (12) is the value of the cash amount invested in the fund at timeT for a perfect hedge in the risk neutral worldQ^γ_λ. That is, if the risk free rate of return was given byδ^γ and the fund was tradeable, the perfect hedging strategy ofλput options would be given by the integrand in (12). Thus we have a quantification of the residual risk as the accumulated value of the perfect hedge with respect to the risk processRt. 1.2 Simulating the model

I have used Matlab to describe the model to my computer. To do this in an as effective way as possible, some rewritings, simplifications and assumptions are needed.

First off, we need to descretify the model presented above. For instance, we have that

Z T 0

φ_tdB_t = lim

n→∞

n

X

i=1

φ_t_i∆B_t_i

≈

n

X

i=1

φti∆Bti

where 0 ≤ t1 < . . . < tn ≤ T and the approximation is better for larger n.

The new and discrete model is updated daily in stead of continuously, which means thatn=T and ∆Bti ∼N(0,1). This gives an error with respect to the continuous model, but it can be argued that this error is insignificant compared to other errors such as parameter insecurity. Also, a portfolio can not be continuously rebalanced since that would cause transaction costs to be enormous.

One could also say that the market is not continuous, and therefore our model must be descretified to better fit the way that the market is behaving. Even daily updating may be a bit too often, but that is what I have decided to use.

One could find an estimate of the error implied by descretifying the model in such a way by adjusting the time steps. This will not be included in this paper.

Secondly, we need to compute∂yp^γ_λ(t, Y(t)) for p^γ_λ(t, Y(t)) as defined in equation (3) on page 4. Here ∂_y is a shorthand notation for _∂y^∂ and will be used consequently.

∂yp^γ_λ(t, Y(t)) = ∂y

e^−r(T^−t)ln(w(t, y)) γ(1−ρ²)

= e^−r(T^−t)∂_yln(w(t, y)) γ(1−ρ²)

= e^−r(T^−t) ∂_yw(t, y) w(t, y)γ(1−ρ²)

Using the above rewritings, one can find a direct derivation with respect to y

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of w(t, y) as

∂_yw(t, y) = ∂_yE_Qh expn

λγ(1−ρ²)(K−Y_T^t,y)⁺oi

= EQ

h

∂yexp n

λγ(1−ρ²)(K−Y_T^t,y)⁺ oi

= E_Q

expn

λγ(1−ρ²)(K−Y_T^t,y)⁺o

λγ(1−ρ²)∂_y

K−Y_T^t,y+ where

∂_y

K−Y_T^t,y+

=

−Y_t^t,1 , K > Y_T^t,y

0 , else

Although this method gives seemingly nice results, the above derivation of w might be a bit dodgy. This is because a function needs to be smooth in all points to be differentiated, which w is not. Actually,w will have a breakpoint almost surely since Y_T^t,y =K with probability 1 for some t. Hence, the derivation should rather be done using Malliavin calculus⁵.

Do also note that for modeling purposes our processY_T^t,y as defined in equation (6) on page 5 may be replaced by

X_T^t,x = x·exp

(δ−η²

2 )(T−t) +η(W_T −W_t)

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= x·exp

(δ−η²

2 )(T−t) +ηp

(T −t)

for ∼N(0,1) without any loss of generality. This is because ρW_t¹+p

1−ρ²W_t² ∼ N(0, t) which we recognize as the distribution of a single Brownian motion.

With this in mind, it can be shown that ∂_xw(t, x) can be expressed as

∂

∂xw(t, x) = E

expn

λγ(1−ρ²)(K−X_T^t,x)⁺o B_T xηT

(15) using the techniques presented in page 57 in di Nunno et al.[8] and the simpli- ficationX_T^t,x ofY_T^t,y as stated in equation (14) above.

1.3 Example

Now, we can look at some examples on how the price behaves for different inputs. Especially, we will look at how the risk aversion affect the price in time, t∈[0,252], and correlation,ρ∈[−0.99,0.99].

The figures showing the price for each set of parameters will contain a total of 6 subplots each. Subplot 1 is a surface plot of the price over both time and correlation while the others are borders of the plot, except subplot 5 which is

5I thank Frank Proske for pointing this out for me

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a cross section of the price over time with correlation fixed at zero. Hence, the five smaller plots are just to help read the surface better. The dots above and below the price lines in the last five plots of each figure indicates the standard deviation of the price when it is calculated 20 times. As we can see from all the figures below, the simulation error is insignificant for small values ofγ. If, on the other hand, γ is as high as 0.5, uncertainty is larger, and hence our simulation error is bigger, as seen in figure 3.

Lets start with the market given in table 1 and suppose the rest of the parameters are as given in the same table.

r= 0.02 g= 0.035 Y₀= 100 µ= 0.06 ν= 0.05 λ= 1 σ= 0.10 η= 0.07 γ = 10⁻¹¹ Table 1: Parameters for the first example

Figure 1: The price with γ ≈0

As we can see from figure 1, the minimum price when risk aversion is close to zero is obtained if it is possible to invest in an asset having the opposite behav- ior compared to fund, in the meaning of correlation being equal to−1. It seems as if correlation plays no role at maturity as we can see from subplot 3. Further, the price is approximately constant over time if the risky asset is independent of the fund, i.e. ρ= 0, as seen from subplot 5. Since the price is approximately linear in time, one can easily calculate the approximate daily change in price

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p(t+∆t)−p(t)

∆t , which is seen to be an increase by 0.33% forρ≈ −1 and a decrease of 0.56% for ρ ≈ 1. It is very close to zero for ρ = 0. The correlation giving the highest price is ρ≈1 at initial time, and ρ = 0.91 at maturity. While the observation for the initial time is ’spot on’, the observation at maturity is not.

The top point at maturity is decided by chance, and the ’real’ price at this time is a flat line.

To conclude one can say that for a low risk aversion, the correlation parameter is of great impact at the initial time. This impact, however, decays as time tends to maturity, as seen in subplots 4 and 6 of figure 1, and is of no relevance at maturity as seen in subplot 5 of the same figure. The dots, representing the standard deviation of the price, are very close to the line, representing the mean of the price, indicating that the price is accurate at 10,000 simulations.

If we increase the risk aversion to γ = 0.1, we can first note that theρ giving the largest price has changed from close to 1 to about 0.75 for the initial time, while it has stabilized around zero for maturity time, as seen in figure 2.

Figure 2: The price with γ= 0.1

Subplots 4 and 6 are quite alike their corresponding subplots in figure 1, while subplot 5 has noticeable higher values. This means that in order to maintain a low price with higher risk aversion, it is crucial to procure a negative correlated asset to hedge in. At least at initial time. When one get closer to maturity, it is also possible to hedge in a positively correlated asset. The main point is that it is not independent. The form of subplot 5 is still approximately the same as it was in figure 1, allthough it is decreasing slightly more here. Accuracy is still high in all subplots agreed by the low standard deviation. Subplots 2 and 3 are

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the ones that have changed the most, Subplot 3 has become more concave than it was when γ was close to zero and subplot 2 has gone from being convex to being mostly⁶ concave.

Further increasing the risk aversion to γ= 0.5, imply a new shape of the price as seen in figure 3. The price is now much more symmetric in the sense that the top point of the price has moved further against zero. The maximum price is also much larger and continues to be over time, as we can see from subplot 5 of figure 3. By increasing the risk aversion from 0.1 to 0.5, the maximum price is increased from about 3.5 to about 10. The price in the extreme cases ofρ being near -1 or 1, have not changed all that much from whenγ was close to zero. It is also important to notice that the uncertainty of the price is starting to show in the interval whereρis small, that is forρ∈[−0.25,0.25] approximately. The uncertainty is largest forρ= 0, where the standard deviation is about 0.8.

Figure 3: The price with γ = 0.5

If one were to raise risk aversion even further, say to γ = 1, one would see ex- actly the same changes as from raising it from 0.1 to 0.5, except one. It seems as if the ρ giving the maximum price stabilizes at about 0.20-0.30. It makes perfect sense that the correlation giving the largest price is positive because one should benefit from diversification, which is done by hedging in a stock with negative correlation. Hence the price should always be larger when hedging in positively correlated stocks. This might be interesting for someone worry- ing that the correlation might change over time, which it easily could. If, for

6It seems as if it turns convex for values ofρclose to -1.

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instance, they knew that they had the ρ giving the highest price, they would know that any change in that correlation, would give a lower price. This could be worth knowing. First I wanted to find thisρ analytical, but I found it to be, well, rather troublesome, so instead I ’solved’ it using Monte Carlo.

r = 0.035 g= 0.00 Y0= 100 µ= 0.07 ν = 0.08 λ= 1 σ = 0.12 η= 0.15 γ ∈(0,0.5]

Table 2: Parameters for the second example

In figure 4, the program has been run 150 times, each time simulating the price with 10,000 paths. The risk aversion γ ranges from close to zero to zero point five with increments of 0.01. This gave me 150 simulations of the ρ giving the largest price for each point inγ, of which I calculated the mean and standard deviation as plotted for both initial time and capital time. Capital time is 1 year, or 252 days, from initial time, while the other parameters are as given in table 2.

Figure 4: Theρ giving the maximum price against risk aversion γ∈(0,0.5]

As we can see from figure 4, the ρgiving the largest price is close to 1 for small γ’s and decrease as γ increase. The decrease, however, does seem to stagger at about ρ = 0.1. The function has decreased to ρ = 0.1 at γ = 0.22, and at γ = 0.5 it is still not significantly lower than 0.1 if one take uncertainty into account. The uncertainty, here represented by the standard deviation of 150 results, on the other hand seem to increase as γ increase. As we can see from the second plot on figure 4, theρ giving the largest price at maturity, is ρ= 0.

The standard deviation is at this time quite large for small γ’s and decreases

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up to γ = 0.14 or so, where it starts to increase to about the same level as for initial time. The reason for the large uncertainty for smallerγ’s at capital time was seen in subplot 3 of figure 1, where the price for a smallγ at maturity is flat, implying that a maximum price could occur over the whole scale due to randomness.

One may further be interested in how this plot looks like for larger risk aversions. The market in figure 5 is the same as the one behind figure 4 and is given in table 2 except that here we letγ range from close to zero to 5. As we saw in figure 4, ˆρ, the ρ giving the maximum price, decreased as γ increased from zero to zero point five. It might look as if ˆρ stabilized at some positive level, or one could think that ˆρ converged to zero. As we can see from figure 5, neither of those is what is really happening. In this figure we can clearly see that ˆρ actually start to increase at γ ≈ 0.6. It is important to note that the uncertainty in this region is quite massive, meaning that the point of where ˆρ starts to increase may well be a bit higher or lower in reality.

Figure 5: Theρ giving the maximum price against risk aversionγ ∈(0,5]

To put some numbers with this, I denoted the mean and standard deviation of ˆ

ρ. These numbers are noted in table 3. As we can see in this table as well as in figure 5, the standard deviation has a steady increase with respect toγ, while the mean has a large decrease at first and then starts to increase again.

It is also worth noticing that the distribution of this ˆρ is quite skew. We can see this by looking at the standard deviation and the 95% confidence interval of ˆρ also plotted in figure 5. Here we can see that the upper 95% quantile is quite close to the standard deviation added to the mean while the 5% quantile is a good deal lower than the standard deviation subtracted from the mean.

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γ 0 0.2 0.4 0.6 0.8 1 E[ ˆρ], std( ˆρ) 0.99, 0 0.14, 0.04 0.08, 0.09 0.08, 0.13 0.09, 0.18 0.10, 0.21

Table 3: The values of figure 5 denoted (γ∈(0,1])

In other words, ˆρ can vary from -0.5 to 0.5, but it is more common that it is on the positive side, hence the mean is positive. Theoretically, it could also be of interest to look at the limit of ˆρ when γ tends to infinity. However, one could argue that no company will have a risk aversion that large, so it would be of limited practical interest. It might also be interesting to note that ˆρ’s distribution is more heavytailed than a normal distribution since its confidence interval is larger than its standard deviation.

Another property worth noticing and which is visualized in figures 1, 2 and 3 is that for all times, all correlations and all risk aversions, the price is growing with respect toρ∈(−1,0). In fact, it can be proved⁷ thatp^γ_λ(t, y) is increasing with respect to ρ∈(−1,0) if ^µ−r_δ >0 and that it is decreasing with respect to ρ∈(0,1) if ^µ−r_δ <0. This is proved in appendix A.1, and it tells us that if the expected growth of the stock is larger than a banks interest rate, which is fairly common, one can obtain a lower price by investing in a lower correlated stock.

If the bank rate should exceed the expected return of the stock, the lowest price would be obtained by having a correlation as close to one as possible.

As we can see from some of the plots in this section, p^γ_λ(t, y) is increasing when µ ≥r for ρ ∈(−1,1) for low γ’s and decreasing when µ≥r for ρ ∈ [0,1) for higher γ’s. This is somewhat harder to prove analytically and has not been emphasized.

1.4 Summary of the model

In this chapter, the model of which we will work with in the rest of this paper has been defined and tested. The main definitions would be the formulation of the utility indifference price of a interest rate guarantee, or put option, and the residual risk of issuing such a guarantee or option. Further, this model was descretified in order to be simulated and lastly, we looked at some examples of the price, showing how the minimum price was obtained for minimal γ’s and how the price increased with γ.

In particular we looked at which ρgave the maximum price asγ was increased and we found that this ρ was positive in expectation, had its maximum at γ close to zero and its local, and probably also global, expected minimum at γ ≈0.6.

7I thank Frank Proske for his help here.

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2 Financial Risk and how to Measure it

Risk is a term most people are familiar with. However, there are several types of risk including market risk, model risk, credit risk and operational risk. The latter is the most recent notion.

2.1 Financial risk

Operational risk is defined by the Basel Committee as the risk of loss result- ing from inadequate or failed internal processes, people and systems, or from external events, as stated in Embrechts et al.[10]. Examples of this risk are technological failure, errors in data processing, fraud, environmental risks, etc.

Market riskis the risk that the value of an investment will change due to changes in the market risk factors, such as interest rates, exchange rates, volatility, correlation, etc.

Credit risk is the risk of financial losses due to the counter party defaulting on a contract, typically a bond-holder being concerned that the bond-issuer will default. The horror example here is the LTCM scandal in 1998. LTCM, or Long Term Capital Management, was founded in 1994 and had amazing returns the first years, but folded in 2000 due to Russia defaulting on a rather large contract in 1998⁸. A good paper for modeling credit risk is Duffie[5].

Model risk can be defined as the risk that a financial institution incurs losses because its risk-management models are misspecified or because some of the assumptions required are not met. For instance, we might work with a normal distribution to model losses, whereas the real distribution is heavy-tailed, or we might fail to recognize the presence of volatility clustering or tail dependences.

Since any financial model is a simplification and therefore an imperfect representation of the economic world, it is fair to say that every risk-management model is subject to model risk of some extent.

Until recent years, the banking and insurance industry only focused on market and credit risk, not spending too much thought on operational risk and the potential losses it could cause. Before, operational risk was included in credit risk, making credit risk more difficult to model than it really was. Now, we want to divide credit and operational risk by implementing the Basel II and Solvency II frameworks. Basel II and Solvency II also requires formal risk modeling for banks and insurance companies, respectively. Model risk has always been a problem and might have increased due to several new and sophisticated but not necessarily consistent models

8I am not saying that Russia is the one to blame for the collapse of LTCM, it would probably have happened sooner or later despite what happened with Russia.

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Basel II⁹ requires that banks are to set aside at least 8% of the total capital invested in risky assets. The aim for Basel II is to ensure that capital alloca- tions are more risk sensitive than they have been in the recent past, separating operational risk from credit risk, quantifying both and preventing regulatory arbitrage¹⁰ by aligning economic and regulatory capital.

It is believed that Basel II, by being an international standard, can help pro- tect the international financial system from the types of problems that might arise should a major bank or even a series of banks collapse, as happened this last autumn. In practice, Basel II attempts to accomplish this by setting up rigorous risk and capital management requirements designed to ensure that a bank holds capital reserves appropriate to the risk the bank exposes itself to through its lending and investment practices.

As mentioned above, the insurance business have a similar framework under development, namely Solvency II which is expected to be implemented within 2012. The aim of Solvency II is to implement more deliberated solvency requirements with respect to the risks that companies face, and to deliver a consistent supervisory system that will be implemented all over the European Union.

Solvency II will reduce the probability that insurance companies get into trou- ble by introducing a far more comprehensive framework for risk management for defining required capital levels and to implement procedures to identify, measure, and manage risk levels than Solvency I could offer. As a side effect, Solvency II will most likely improve the confidence among policyholders (both current and potential) that the financial part of the insurance companies are steady by reducing the chances of policyholders losing if insurance companies get into difficulties. Longevity will also be considered in a greater extent then under Solvency I.

It is in my opinion that neither the Basel II nor the Solvency II framework would have prevented the current financial crisis from evolving even if it had been implemented many years ago. However, I do believe that it could have greatly reduced some of the severe negative effects we have seen lately and probably will see more of in 2009.

The field of risk analysis is a rather large one, and I will not attempt to cover it all. Instead, I will from now on focus on market and model risk and how to measure and control it. The primary aim, when modeling risk, is to quantify likely losses of a portfolio. To do this quantification, we need to know what a risk measure is.

9Recommendations on banking laws and regulations issued by the Basel Committee on Banking Supervision

10Arbitrage that can arise from regulated institutions taking advantage of the difference between its economic risk and the regulatory position it has

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2.2 Risk measures

Risk can be thought of as a random variable, say Xt, which is defined on the same filtered probability space as in chapter 1, namely

Ω,F,{F_t}_t∈[0,T_], P

. A risk measure ϑ is then a relation between the set of random variables, Xt, and the real line, that is ϑ(Xt) ∈ R. If, for instance, Xt is the expected loss of a portfolio given some security level,ϑ(X_t) can be the additional amount of money the company need to set aside to survive with that security level. The definition of risk measures is quite general, meaning that not all risk measures necessarily aregood risk measures. Therefor, one might want risk measures to fulfill some additional conditions. For instance, one would probably want a risk measure to not exceed the largest possible loss.

For this reason, one has divided risk measures into several sets and subsets. The largest set is the set of monotary risk measures. This is the set of risk measures that satisfy monotonicity and translational invariance. A subset of these are the convex risk measures, recognized by the above set of economically desirable properties, as well as convexity. In 1999, Artzner et al.[13] performed the first systematic study of risk measures properties within finance and defined the class ofcoherent risk measures. This set is a subset of the convex risk measures satisfy the following axiom:

Axiom

A coherent risk measureϑis a risk measure, which for all risks X_tand Y_t and all constantsc≥0 satisfy

(a) Translational invariance: ϑ(Xt+c) =c+ϑ(Xt) (b) Positive homogeneity: ϑ(c·Xt) =c·ϑ(Xt) (c) Monotonicity: ifX_t≤Y_t, thenϑ(X_t)≤ϑ(Y_t) (d) Subadditivity: ϑ(Xt+Yt)≤ϑ(Xt) +ϑ(Yt)

In words, one could explain these axioms in the following way: Translational invariance: Adding an amount of cash to the portfolio decreases its risk by the same amount. Positive homogeneity: If we increase the size of all risky positions in a portfolio, the risk of the portfolio will be increased by the same size. Monotonicity: If losses in one portfolio are larger than losses in another portfolio for all possible risk scenarios, then the risk of the first portfolio is higher than the risk of the second. Subadditivity: The risk of a portfolio is smaller than or equal to the sum of risks of its sub portfolios, or in other words;

Risk in general should be reduced by diversification.

There exists numerous versions of this axiom depending on whether, for instance, one define losses as negative or positive values. In our case, X is the value of a risky position at the end of the holding period, hence it is a random variable at one point in time, namely the terminal time. The risk in this paper

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is denoted byR^γ_λ and is defined on page 6.

Now that we have defined what the risk is, lets have a look at some of the ways to measure it.

2.2.1 Standard deviation

In 1952, Harry Markowitz wrote a paper about modern portfolio theory where he defined the standard deviation of the value of a portfolio as a risk measure.

This was the very first widely used risk measure. While standard deviation is quite easy to calculate, it penalizes losses as well as profits, and is therefor, in general, not a very good risk measure. It is subadditive, but still not coherent since it violates the monotonicity property.

2.2.2 Value at Risk

Value at Risk (VaR) has been the most used risk measure since J. P. Morgan released their RiskMetrics system in 1994. VaR gives us the answer to how much the value of a portfolio can drop given some probability level α. We say that VaR is a downside risk measure, since it typically describes the probability boundary of potential losses. One can define Value at Risk formally in the following way:

Definition 2.1. Given a risk X with cumulative distribution function F_X and a probability level of α∈(0,1), then

V aR_α(X) = F_X⁻¹(α) =inf{x∈R:F_X(x)≥α}

is the Value at Risk of X at α level.

α is often chosen to be among 0.95, 0.99 or 0.999. From a statisticians point of view, VaR is nothing but the α quantile of some sorted distribution, and says that with a confidence ofα·100%, you will not loose more than VaR_α. In Jorion[14] page 27, VaR is represented as V aR_α(X) = E[F(X)]−Q_α[F(X)]

where Qα[F(X)] is the quantile of the qumulative distribution ofX, matching the confidence level α. This representation is nothing but VaR viewed from a different angle and is consistent with the definition above.

Even though VaR has become the benchmark risk measure in the financial world, it is not perfect. As almost everything else in this world it has some flaws that one should know before using it. Being a one-period risk measure, VaR does not care what happens with the portfolio value in the holding period, it only cares about the level at capital time. It also assumes that the market position is the same at all times, which is highly unlikely in practice. However, one can argue that these flaws are minor and will drown in comparison to other uncertainties.

More serious is the fact that VaR does not measure the potential size of a loss, given that it exceeds VaR. The most serious of VaR’s flaws, is the fact that it

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Bonds Initial Defaults Risk measure No Soft A Hard A Soft B Hard B VaR0.95 ES0.95

A 104.6 108 100 0 108 108 4.6 64.6

B 104.6 108 108 108 100 0 4.6 64.6

A+B 209.2 216 208 108 208 108 101.2 101.2

Probabilities: 0.9 0.02 0.03 0.02 0.03

Table 4: Table showing how Value at Risk is not subadditive, while Expected Shortfall is.

is, in general, not subadditive, and hence not a coherent risk measure. For the normal distribution and other light-tailed distributions, VaR might well be subadditive. It is when one is dealing with either extremely skewed distributions (e.g. exponential), heavy tailed distributions (e.g. Pareto) or distributions with a special dependence structure (e.g. copulas) that one has to pay attention. The implications of not having a subadditive risk measure are severe when modeling financial or economic values, as such a risk measure dissuade diversification.

One can then create severe aggregation problems when adding risk.

VaR has been popularized as the risk measure of choice among investment banks wishing to measure their portfolio risk for the benefit of banking regu- lators. However, due to the lack of subadditivity, VaR appears to be unfit for such calculations.

We can now take a look at a some examples of using VaR, beginning with a practical example of how subadditivity may fail to work.

Example 2.2. Suppose we have a bond A and that, at maturity, there are three possible outcomes, i.e. Ω ={ω₁, ω2, ω3}

ω₁ No default: The bond redeems both its face value of 100 Euro and the coupon of 8 Euro with probabilityP(ω₁) = 0.95

ω1 Soft default: The bond redeems only its face value of 100 Euro with probabilityP(ω₂) = 0.02

ω₁ Hard default: The bond redeems nothing with probabilityP(ω₃) = 0.03 Note: If we ignore the second possible outcome, and say that P(ω1) = 0.97 and P(ω₃) = 0.03, we get that VaR_0.95(A) = 0, even though the risk of such a bond is greater than zero.

Further, suppose there is another bondB that is identical toA, but issued by another agent. The risks of these two bonds are each others opposites, in the meaning that if bondA defaults, bondB will not, and vice versa.

As we can clearly see from the second last line of table 4, VaR is not subadditive since 4.6 + 4.6<101.2, hence VaR clearly disfavors diversification.

To end my discussion of VaR, I would like to take a look at an example found in Embrechts et al.[10] showing that VaRis subadditive for Gaussian distributed risks.

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Example 2.3. SupposeX₁ andX₂ are jointly Gaussian distributed with mean µ and covariance matrix Σ

µ= µ1

µ₂

and Σ =

σ₁² ρσ1σ2

ρσ₁σ₂ σ²₂

where ρ∈[−1,1] andσi >0, i∈ {1,2}. Ifα∈[0.5,1), then

VaR_α(X₁+X₂)≤VaR_α(X₁) + VaR_α(X₂) (16) Since (X₁, X₂) is bivariate Gaussian distributed, X₁, X₂ and X₁+X₂ are all univariate Gaussian distributed. It follows that

VaRα(Xi) = µi+σiqα(), i∈ {1,2}

VaR_α(X₁+X₂) = µ₁+µ₂+ q

σ₁²+ 2ρσ₁σ₂+σ²₂q_α()

where qα() is theα-quantile of a standard Gaussian distributed random variable, . Inserting this into (16) gives us that σ²₁ + 2ρσ1σ2 +σ₂² ≤ (σ1 +σ2)² sinceρ≤1. Hence the subadditivity property is kept.

It is shown in McNeil et al.[1], Theorem 6.8, that VaR is subadditive for the set of linear combinations of components of a multivariate elliptical distribution.

2.2.3 Expected Shortfall

Since VaR does not tell us anything about the size of our potential loss, Artzner et al.[13] considered the notion of Expected Shortfall (ES) also known as conditional tail expectation. In words, one can define Expected Shortfall at α-level as the expected risk in the worst α·100% of the cases. The meaning of this level is that ES ignores the most profitable but unlikely possibilities for high α’s while it focuses on the worst losses for lowα’s. In more formal terms, one can define ES as follows:

Definition 2.4. Let X be a risk and α ∈ (0,1). Expected Shortfall is then defined as the conditional expected risk given that the risk exceeds VaRα(X):

ESα(X) =E[X|X >VaRα(X)]

Note: The direction of the inequality sign in the definition above is decided by how you define VaR. The point is that the expectation is to be taken over the variables that exceed VaR in the tail. A more precise representation is made by McNeil et al. in [1] page 44 by observing that for a continuous random variable X, one has that ∀α∈(0,1)

ES_α(X) = 1 1−α

Z 1 α

VaR_u(X)du (17)

For such continuous X, Expected Shortfall is subadditive and therefor a coherent risk measure, see Artzner et al.[13]. For discrete random variables X,

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equation (17) needs to be slightly modified to achieve the same properties as in the continuous case. For a discrete random variableX, one has∀ α∈(0,1)

ESα(X) = 1

1−α(E[X|X≥VaRα(X)]

+ VaRα(1−α−P(X≥VaRα(X)))) (18)

= lim

n→∞

P[n(1−α)]

i=1 X_i,n

n(1−α) a.s. (19)

Here, in equation (19), X1,n ≥ . . . ≥ Xn,n is the ordered statistics of the se- quence (Xi)i∈N of independent identical distributed (i.i.d.) random variables and [n(1−α)] denotes the largest integer not exceeding n(1−α). A proof of this latter equation is found in Proposition 4.1 of Acerbi and Tasche [4].

Among ES’s properties we find that ES_α increases asα increases and that ES_α is worse than, or equal to, the Value at Risk (VaR_α) atα level, i.e. ES is more conservative. See example 2.3 for a case where ES is subadditive while VaR is not.

2.2.4 Conditional Value at Risk

A third highly popular risk measure is Conditional Value at Risk (CVaR) also known as ’Mean Excess Loss’, ’Mean Shortfall’ or ’Tail VaR’. CVaR is meant to be an extension of VaR in the sense that it coincides with VaR for elliptical distributed risks, but remains coherent for general distributed risks. Conditional Value at Risk is computed by assessing the likelihood (at a given confidence level) that a specific loss will exceed the Value at Risk. Mathematically speak- ing, CVaR is derived by taking a weighted average between the Value at Risk and losses exceeding the Value at Risk, as we shall see soon. The following definition of CVaR forcontinuous X is found in Rockafellar and Uryasev[15]:

Definition 2.5. Conditional Value at Risk at level α ∈ (0,1] of a continuous random variableX is:

CVaRα(X) =E[Ψα(X, ζ)]

where Ψα(X,·) is the α-tail distribution of X defined as, Ψα(X, ζ) =

(Ψ(X, ζ)−α)/(1−α) , ζ ≥VaRα(X)

0 , else

andΨ(X, ζ) =P(X|X≤ζ).

Note that both Ψ and Ψ_α are non-decreasing and right-continuous, and that Ψα(X, ζ)→1 asζ → ∞. Hence, theα-tail distribution referred to above is well defined. As a side-note,ζ can be interpreted as VaR and in fact, for continuous distributed risks, CVaR and ES coincide.

However, since I am interested in evaluating simulated distributions, i.e. not continuous ones, I will need a representation of CVaR that is more appropriate.

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Luckily, Rockafellar and Uryasev[15], in their Proposition 6, has proved the following.

Proposition 2.6. Let λα(X) be the probability assigned to the loss amount VaR_α(X) by theα-tail distribution, that is

λ_α(X) = Ψ(X, V aR_α(X))−α

1−α ∈[0,1]

whenΨ(X, V aRα(X))<1,

CV aR_α(X) =λ_α(X)·V aR_α(X) + (1−λ_α(X))·ES_α(X) while Ψ(X, V aRα(X)) = 1gives

CV aR_α(X) =V aR_α(X)

The proof of this and the reason behind the choice of λ_α is included in Rock- afellar and Uryasev[15].

While the former proposition is best for understanding, the next one is probably better for computations.

Proposition 2.7. Suppose the probability measure P is concentrated in finitely many pointsy_kof Y such that for eachx∈X, the distribution of lossz=f(x, y) is likewise concentrated in finitely many points, and Ψ(x,·) is a step function with jumps at those points. Fixing x, let those corresponding loss points be ordered as z₁ < z₂ < . . . < z_N, with P(z_k) = p_k > 0. Further let k_α be the unique index such that

kα

X

k=1

pk≥α >

kα−1

X

k=1

pk

The VaRα of the loss is then given by

VaR_α(x) =z_k_α while the CVaR_α is given by

CVaR_α = 1 1−α





kα

X

k=1

p_k−α

! z_k_α+

N

X

k=kα+1

p_kz_k





The proof of this is also carried out in Rockafellar and Uryasev[15] under their Proposition 8. I suggest taking a look at this paper for further reading about this risk measure.

It is this last proposition I have used to calculate CVaR in my programs.

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2.2.5 Entropic risk measure

The scientific meaning of an entropy is a measure of the disorder, or uncertainty, in a given system. An entropy is a measure on how great the unexpected changes in a system is. The entropic risk measure is said to be the most famous example of a convex risk measure, introduced by F¨ollmer and Schied[9]. The definition is quite straight-forward and is given by the following

Definition 2.8. The entropic risk measure, ENT, of a non-negative random variable X is given by

ENT(X) = 1

γ logE[e^−γX]

where γ is the risk aversion introduced in the first chapter of this paper.

The entropic risk measure is in general not coherent but it is convex. By conditioning the expectation on a filtration, F_t generated by the Brownian motion, the entropic risk measure can be extended to a dynamic risk measure as defined in section 3 of Barrieu and Karoui[11]. I am confident that the field of dynamic risk measures is a very interesting and useful one, but far to extensive to be treated in this short paper. I refer to Barrieu and Karoui[11] for a more comprehensive talk about the subject. As it turns out, the risks in this paper are not non-negative, so this measure should not be used here. It is still an interesting measure having several areas of use and for this, I will include it in my calculations.

2.2.6 Other risk measures

There are of course many other risk measures than the ones I have looked at here including theIncremental risk measures (s.a. Incremental VaR (IVaR), Incre- mental Expected Shortfall (IES), and Incremental Standard Deviation (ISD)), Marginal Value at Risk and Conditional Drawdown at Risk. This paper, however, is not meant to be a total run-through over all existing risk measures.

It is rather meant to be an examination of how some risk measures behaves when used on utility based pricing of interest guarantees and the risk of such.

I will for these reasons not go into close examination of rare and elusive risk measures, albeit I would say it is needed.

2.2.7 Important purposes of risk measures

As I mentioned earlier in this chapter, the primary aim of a risk measure is to quantify likely losses or the likeliness of a given loss. This is a quite general statement, and I would like to mention some of the most important purposes of a risk measure.

Determination of risk capital and capital adequacy. The principal function of risk management within finance, included insurance, is to determine the amount of capital a financial institution needs to hold as a buffer against unexpected

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losses in its portfolio in order to satisfy a regulator concerned with the solvency of the institution.

Insurance premiums compensate an insurance company for bearing the risk of the insured claims. The size of this compensation can be viewed as a measure of the size of these risks.

Management tool. Risk measures are often used as a tool for limiting the amount of risk a unit within a firm may take. For instance, traders in a bank are often constrained by the rule that the daily 95% Value at Risk of their position should not exceed a given bound.

2.3 Summary of risk measures

In this chapter we have looked at several different risk measures and their pros and cons. After working with them all, I would like to emphasize Expected Shortfall for its properties and its easy-to-understand calculations and meaning. In the end of the day, what we really want from such a measure, is for it to be easy to use and easy to explain for others.

I would also like to note that even though the entropic risk measure need positive inputs by definition, and the residual risk defined in chapter 1 might well be negative, this risk measure will still be evaluated and used later in this paper.

Strictly out of curiosity.

The difference of CVaR and ES. When doing research for this part of my paper, I found that there exist some inconsistent opinions on CVaR and ES. This inconsistency consist mainly of some people considering ES and CVaR to coincide, while others do not. After careful reading of selected opinions on the subject, I decided to agree with the ones saying there is a difference even if it is rather small. In this paper, the difference of ES and CVaR is that CVaR is a weighted average of VaR and ES, as defined earlier in this chapter. The difference in value might not be significant, but the two ideas are quite inequal.

As a rule of thumb, one can say that risk measures represented by an expectation are in general coherent. The entropic risk measure is not included in this rule.

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3 The Relationship of Price and Risk

In this chapter, I will give a close examination of the relationship between the utility indifference price, p^γ_λ, and the residual risk, R^γ_λ, as defined in chapter 1. We rememberR^γ_λ to be the risk a company is left with after hedging. This examination will be done by looking at the price and risk of different sets of parameters given in tables below. I will also check for which sets of parameters the coherence property of the different risk measures is retained. The funds initial value,Y₀, will be equal to 100 in all the examples below.

All calculations in this chapter are done by double Monte Carlo with 10,000 simulations repeated 20 times.

3.1 A ’money back’ guarantee

This example is fetched from Benth and Proske[6]. Consider a contract agreed at time t ∈ [0, T] which guarantees that an investor gets his money back at a predefined time, T. In other words, the guaranteed rate of return is zero percent. This is a relevant situation for Norwegian pension funds, where the investor may have buffers to cover possible deficits in the fund. This buffer is built up in years with surplus returns over the guaranteed, which is usually around 3.5%. However, new legislations enforce the manager to cover a possible negative return on the fund irrespective to the amount of buffer capital at hand. Since most investors have large buffers, the manager is basically issuing an at-the-money put option, i.e. a guarantee against a negative return.

Suppose that the risk free rate of return,ris equal to 3.5%, the expected return of the pension fund, ν = 8% and its volatility, η = 15%. The hedging asset S has a slightly lower expected return, µ = 7% and its volatility, σ = 0.12 indicates less risk. These are all yearly values, and are summarized in table 5 with the number of options issued,λand the risk aversionγ.

r= 0.035 g= 0.0 p^B&S = 4.32 µ= 0.07 ν= 0.08 λ= 1 σ= 0.12 η= 0.15 γ = 10⁻¹¹

Table 5: Money back guarantee with quite high drift and volatility Lets first look at the minimum price obtained whenever the insurance company is tolerant to risk, i.e. they have a low risk aversion. As we can see in figure 6, the lowest price for this market is given at (t, ρ) = (0,−1) and is 1.87.

This is significantly lower than the price found at (t, ρ) = (0,1), which is 4.27.

Note that both are lower than the B&S price. As time goes by, the price will increase or decrease depending on whetherρis negative or positive, respectively.

For ρ = 0, the price seems to remain roughly the same for all times and is approximately equal to the price at time T for allρ. This is a little deceptive since if one were to zoom in on it, one would see that this is not the case. It

The price and risk of guaranteed interest rate products